Shotgun pellet loads and infliction rates in pink-footedgeese Anser brachyrhynchus

Authors: Noer, Henning, and Madsen, JesperSource: Wildlife Biology, 2(3) : 65-73

Published By: Nordic Board for Wildlife ResearchURL: https://doi.org/10.2981/wlb.1996.034

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ORIGINAL ARTICLES

Shotgun pellet loads and infliction rates in pink-footed geese Anser brachyrhynchus

Henning Noer & Jesper Madsen

Noer, H. & Madsen, J. 1996: Shotgun pellet loads and infliction rates in pink-footed geese Anser brachyrhynchus. - Wild], Biol. 2: 65-73.

Pink-footed geese Anser brachyrhynchus caught in western Denmark during March- April 1990-1992 were X-rayed to detect shotgun pellets. Among first-year and older geese, pellets were detected in 25% (N = 69) and 36% (N = 286), respectively. A sim­ple theoretical model is proposed, relating frequencies of pellet carriers in different age classes, adult survival and the annual rate at which pellets are inflicted upon geese. This model resulted in estimates o f pellets being inflicted upon 1,000 first-year and 800 older geese annually. However, annual bag is ca 1,000 first-year and 2,000 older geese, and thus, expectedly, pellets should be inflicted upon approximately two older geese for each first-year. From this it is concluded that the model leads to unrealistic estimates because the observed frequencies of first-year and older pellet carriers are inconsistent; if 25% of the geese carry pellets after their first hunting season, more than the observed 36% of the older geese should be carriers. One explanation of this could be statistical uncertainty of the observed frequency of first-year carriers. A frequency of 15% pellet carriers (lower 95% c.l.) after the first hunting season could account for the inconsistency. As an alternative explanation, lower survival of pellet carriers is pro­posed. A more general model including differential mortality predicts that survival rates o f 87% and 78% for non-carriers and carriers, respectively, could explain the ob­served carrier frequencies. Use of lower 95% c.l. leads to estimates of pellets being in­flicted upon a minimum of ca 0.5 goose per bagged one. If differential survival exists, this ratio will be somewhat higher.

Keywords: pink-footed geese, shotgun pellets, X-ray analysis, hunting

Henning Noer & Jesper Madsen, National Environmental Research Institute, Depart­ment o f Coastal Zone Ecology, Kal0, Grenaavej 12, DK-8410 R0nde, Denmark

Received 29 May 1996, accepted 12 September 1996

Associate Editor: Nigel G. Yoccoz

Amongst hunted goose species, X-ray investigations have demonstrated frequencies of adult individuals car­rying shotgun pellets ranging between 28% and 62% (Elder 1955a,b, Grieb 1970, Ankney 1975, Jonsson et al. 1985). For pink-footed geese Anser brachyrhynchus, Eld­er (1955a) found 39% adult pellet carriers among individ­uals belonging to the Iceland/Greenland population, caught on wintering grounds in the British Isles.

Though these frequencies appear high, they are not ne-

© WILDLIFE BIOLOGY

cessarily evidence that pellets are inflicted upon large numbers of geese within any single hunting season. Geese are long-lived, and relatively small annual rates of inflic­tion could thus eventually lead to high frequencies of pel­let carriers. Therefore, assessment of the number of indi­viduals hit by shot and surviving as pellet carriers each year is essential for a proper interpretation of the frequen­cies measured by X-ray photography. Moreover, if these rates can be assessed and reliable data on bag statistics

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and population sizes are available, the rate at which pel­lets are inflicted can be compared to harvest rates, in or­der to assess the ratio of wounded to killed birds. Because this ratio would provide a measure of hunters' perform­ance, it should be of interest to both scientists, wildlife administrators and hunters,

In order to obtain data for the Svalbard population of pink-footed geese, one of us (JM) initiated X-ray record­ings in 1990. After an initial period of data compilation, a simple theoretical model was developed in order to as­sess the 'annual rate of infliction' - defined as the fraction of the population that is hit by shot and survive as pellet carriers each year. However, application of this model led to the realisation that the observed frequencies of juve­nile and older carriers observed by X-ray photography did not correspond. Accordingly, the purposes of this paper are: 1) to present the results of X-ray investigations of the Svalbard population of pink-footed geese, 2) to present a simple model intended for assessment of annual infliction rates, 3) to compare the results to the bag statistics for the Svalbard population of pink-footed geese, thereby dem­onstrating that the observed frequencies of first-year and older pellet carriers are inconsistent in relation to the age distribution of the bag, and 4) to discuss the possible causes of this inconsistency.

Material and methods

Data collectionGeese were caught by cannon-net at Vest Stadil Fjord (56°12'N, 08°08'E), Western Jutland, Denmark, 27 March 1990,3 April 1991 and 25-26 March 1992. An ad­ditional sample consisting of 11 geese collected during a test catch in 1988 was included. Ageing and sexing was based on plumage characteristics and cloacal examina­tion. Pink-footed geese moult into 'adult' plumage during their second summer, which means that two age classes can be discriminated in early spring catches, viz. 'first- year' and 'older'.

The geese belong to the Svalbard breeding population. Autumn migration is via the Norwegian mainland and Western Jutland, from whence they leave for wintering grounds in the Netherlands and Belgium during Novem­ber. Depending on weather conditions, geese may return to Western Jutland from mid-December onwards. The species is protected in Germany, the Netherlands and Bel­gium, while open seasons exist in Norway (including Svalbard) and Denmark. The Danish season extends from 1 September to 31 December (Madsen 1987). Since 1994 hunting has also been permitted 1-15 January, but exclu­sively on the sea territory where very few pink-footed geese are bagged (Madsen et al. 1996). Thus, sampling was done 2.5-3 months after the hunting season closed.

The population size (November census counts 1993/ 1994) is 34,000 individuals, of which 4,500 are first-years (Madsen & Mitchell 1994). The annual bag is ca 3,000 individuals, of which ca 2,000 are taken in Denmark, the rest in Norway (Madsen et al. 1996). Wing surveys show that first-years make up one third of the Danish bag (Clausager 1990, 1991, 1992, 1993, 1994). We therefore assumed total bag composition to be ca 2,000 adults and1,000 juveniles. Approximately half of the Danish bag is taken after the return of the geese from the Netherlands (i.e. after the November counts), and we subtracted 50% of the Danish bag size from the results of the November census in order to estimate numbers of pellet carriers from percentages observed in April. The resulting figures were ca 4.000 first-years and 29,000 older geese.

X-ray photographyX-raying of geese was done during capture sessions, us­ing commercial portable equipment. Geese were masked and wrapped in cloth in order to restrain them during ex­posure. Picture frame size was 45 x 30 cm. and when present, pellets were easily identifiable in the photo­graphs. Two cases where pellets may have resided in the gizzard, i.e. have been ingested and not inflicted, were ex­cluded.

Infliction ratesIf the exact age of captured geese could be determined, estimation of infliction rates could be done by subtraction of carrier frequencies of successive cohorts. However, only two age groups can be discriminated, 'first-years' and 'older'. First-year individuals X-rayed in March-April have experienced only one hunting season, and for these the infliction rate equals the percentage of pellet carriers. For older individuals, the relationship between infliction rate, adult survival and frequency of carriers can be de­rived under certain simplifying assumptions.

Assume that the population is stable, and that annual survival <|) is constant after the first year of life. Thus, sur­vival is identical for carriers and non-carriers. Assume further that the probability of pellets being inflicted upon a goose is constant over years and individuals, but differs between the two age classes. This means that within each hunting season, pellets are inflicted upon all individuals older than one year with equal probability 7t. This defini­tion ignores the possibility that pellets may be inflicted upon some geese more than once within a single season, but since it is likely that very few geese will experience - and even fewer survive - this treatment, the above sim­plification is considered a good approximation.

Under these assumptions, the proportion of 'recruits' (i.e. individuals in their second year) in the 'older' seg-

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ment of the population will be 1 - <f). At the time of re­cruitment (summer moult), this cohort still has experi­enced only one hunting season, and after that some pro­portion 9j was carriers. At the time of X-raying in the fol­lowing spring, these individuals will have survived their second hunting season, and if their survival rate was iden­tical to that of older geese they will still make up a pro­portion 1 - <J) of the geese aged as 'more than one year'. If a proportion 0j carried pellets after the first hunting sea­son, the probability of an individual not being a carrier af­ter two seasons will be

P{#Pellets = 0} = (1 - 0j)( 1 - tc).

For each following age class, the probability of an indi­vidual not carrying pellets at the time of X-raying should therefore be

P { # P elle ts = 0 | A g e = i) = (1 - 0^(1 - Tt)1-1 , ( i= l,2 ,...) .

From this relationship, the expected proportion of non­carriers in the 'older' part of the population (geese aged more than one year) can be calculated by averaging over the age distribution

1 - 9. = (1 -M 1 -0 ;) I d - Tt/V '-1h=\

where h denotes 'adult' age (i.e. full age minus one) and 0a the proportion of pellet carriers in the 'older' segment of the population. The sum of this series is convergent and readily calculated as

1 - 0a(1 - 9,)(1 - <|»(1 ~ n)

1 - <t>(l - 7t)( 1)

Jonsson et al. (1985) used a similar approach to predict percentage of pellet carriers. However, their calculations were based on the assumption that infliction rates are age- independent and thus can be estimated directly by 0, for all age classes. Since first-year individuals are overrepre­sented in the bag amongst virtually all waterfowl species, this assumption is unlikely to be supported by actual data.

According to formula I, the frequency of adult carriers can be explained in terms of frequency of first-year car­riers, annual adult survival and the probability that pel­lets are inflicted upon an adult goose during any one hunt­ing season. In reality, information on 0j; 0a and <|> is avai­lable, while n is unknown. By rearrangement, however, the expression can be solved with respect to n:

(1 - 0,)(1 - 0) + (1 - 0„Xt»In the present case, estimates of 0j and 0a were obtained simultaneously, but other cases could be envisioned where first-year and older geese were caught on separate

occasions. Information on survival was obtained from subsequent resightings of released neck-banded geese (Madsen & Noer 1996). Hence, n is not an estimate de­rived from Maximum-Likelihood theory in the well- known statistical sense, but rather a combination of three different estimates obtained from three different sets of data. The variance of 7t can be assessed either by means of Bootstrapping (i.e. Monte-Carlo simulation of large numbers of data sets having the observed values as 'means'), or by means of Taylor-expansion of formula 2 above, the socalled delta method (Efron 1979, Seber 1982).

The sampling distribution of 7t was found empirically, by generation of data sets with Nj and Na individuals, us­ing the observed values of 0, and 0a as the probabilities that X-rayed individuals carry pellets. Thus, the distribu­tion of jc was simulated by 'Bootstrapping' frequencies of first-year and older carriers from binomial distributions with parameters (Nj; 0j) and (Na; 0a) and values of <j> from a normal distribution with expectation and variance esti­mated from capture-recapture analysis (Madsen & Noer 1996). By use of Taylor expansion, an approximate ex­plicit expression for the variance of 7t is

V[it] - A 2 F[0 ] + A 2 V[0J + A 2 V[<l>] (3)oQj d0a do

where

= (1-9, ,)(1-<|»00,. [(l-0,)(l-<j>) + (1-0JW2

dn_ = ( l - 0 7)(l-0 )

00„ [(1—0_/)(l—<t>) + (1—0a )<|)]2and

dn = (9a- 6 ; ) ( l - 6 fl)a<t> [( i-0 , )(i-<i» + d -0 „ )0 ]2

Formula 3 ignores second-order partial derivatives, par­ticularly the covariance terms between 0,, 0a and <|>. Since the three parameters are estimated from different data sets and so should have zero sampling covariances this as­sumption was considered justified. In particular, identi­cal survival rates of carriers and non-carriers would mean that the covariance between <|) and 0j and 0a, respectively, should be zero.

Moreover, the calculation of 7t by means of formula 2 could potentially be biased, i.e. the expectation of the re­sulting value differs from the true one. Bias was assessed from Taylor-expansion, too, by means of the formula

b “ + - ~ V A R ( Q a) + ~ ^ 4 vA/?((|>) (4)200: ; 2 002 2 0(|>2 V

where

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<?n = 2 (1—8a)(l—(}))230,2 [ ( i - e ,) ( i - 4» + d - e „ )0]2

d^K = (1 —0y )(1 —<t>)<t>

302 [(i-e 7)(i-<t>) + ( i- e a)<))]2and

2 ( e . - e ^ ^ q - e j

302 [d-e,)d-<W + ( i - 0u)<M2Testing of the null hypothesis H0:7t = 0 was done by com­paring 0 and 0a, which can be done for example by a 2 x 2 '/2~test. Since the alternative hypothesis, to be ac­cepted if H0 is rejected, would be H,: 7t > 0 (i.e. 0a > 0,), the test should be one-sided. This could be done by ac­cepting H0 if 0a < 0j and using the 90% quantile - i.e. a significance level 1 -a of 0.05 - as limit of significance for all other results. For a ^-distribution with df = 1, this would correspond to a significance limit of 2.71. If the two frequencies are not significantly different, 7t is not significantly larger than 0.

Results

Frequencies of pellet carriersIncluding the 11 specimens collected in 1988, a total of 355 pink-footed geese were X-rayed (Table 1). Frequen­cies of carriers in each of the three sampling years (ex­cluding 1988) were very similar (see Table 1, first-years X2 = 0.40, df = 2, P > 0.70, older geese x2 = 0.06, df = 2, P = 0.82). Hence, data were pooled over years.

Of 69 first-years, 17 carried pellets (0, = 0.246, see Ta-

Table 1. Numbers of pink-footed geese in the two age classes with and without shotgun pellets, detected by X-ray photography of 355 geese caught using cannon-net at Vest Stadil Fjord, West Jutland, Denmark, during March-April 1988-1992.

First-year Older

Year + Pellets - Pellets + Pellets - Pellets Total

1988 2 2 4 3 111990 1 4 22 41 681991 8 22 47 87 1641992 6 24 30 52 112Total 17 52 103 183 355

ble 1). 'Exact' 95% confidence limits were calculated by interpolation between tabulated values (Anon. 1970) and were [0.151; 0.365]. For older geese, 103 out of 286 were carriers (0a = 0.360, see Table 1). The 95% c.l. were [0.300; 0.407],

Test of the difference between carrier frequencies of juvenile and older geese resulted in a test statistic of x2 = 3.22. Using the one-sided test outlined above, the differ­ence was significant (P = 0.04), though it should be not­ed that if the test is interpreted to be two-sided, P would be ca 0.07 and H0 accepted.

Mean and variance of number of pellets were 1.60 and 0.54 for first-year carriers and 2.45 and 3.74 for older car­riers. The former carried up to three pellets, the latter up to 10. Adult males tended to carry more pellets than fe­males (mean and variance were 2.62 and 4.95 for males, and 2.21 and 1.85 for females). The larger means for adults were caused by a few geese with high numbers of pellets, and variance increased with mean number of pel­lets, which complicated statistical comparisons. Assum­ing that the means were normally distributed, approxi­mate test statistics can be calculated from the standard

e . e a a

TT TT TTFigure 1. Predicted relationship between frequency of adult carriers (9a) and infliction rate (it), for an adult survival rate (|> = 0.838 and a fre­quency of first-year carriers 9j of 0.246 (left diagram). Variation in the predicted function resulting from variation in 9j (upper and lower 95% c.l.) is shown as solid lines in the diagram in the middle and variation in the predicted function resulting from variation in (|> (upper and lower 95% c.l.) is shown as solid lines in the right diagram.

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normal distribution, i.e. the distribution with mean 0 and variance 1. Using this, the difference between older males and females was not significant (z = 1.14, P = 0.13), but may be real since similar tendencies were found for oth­er species by Elder (1955b). The difference between first- years and older geese, however, was significant (z = 2.36, P = 0.01). The latter test should be considered tentative since the 17 first-year carriers included in the mean make up a relatively sparse sample for the test.

Infliction ratesInfliction rate for first-year geese was estimated directly from 0j = 0.246. Based on evidence from resightings of neck-banded individuals, the estimated annual adult sur­vival of pink-footed geese in the Svalbard population is <|> = 0.838, with 95% c.l. [0.815; 0.859] (Madsen & Noer 1996).

For the observed values of 0j and <]), formula 1 predicts that 0a increases sharply for small values of ti. Annual in­fliction rates between 0.0 and 0.1 can account for adult carrier frequencies of up to ca 50% (Fig. 1). Thus, the high frequencies of pellet carriers found by X-raying of hunt­ed geese may be explained by infliction of pellets upon only a few per cent of the population annually.

An initial sensitivity analysis was made by calculating the graphs resulting from the upper and lower 95% c.l. for 0 and (|>. This showed that uncertainty in 0, has a consid­erable impact on the predicted relationship, while uncer­tainty in (]) is less important. The uncertainty of (j> has the largest implications for frequencies of adult carriers be­tween 0.60 and 0.90 (see Fig. 1), but nearly all reported frequencies (studies cited above) lie below this range.

Therefore, uncertainty in the input survival rate is of less practical consequence.

Direct calculation by means of formula 2 leads to a pre­dicted infliction rate of 7t = 0.028. The calculated vari­ance of 7t (formula 3) was V[7t] = 0.2390632-0.002691 + 0.2815672-0.000806 + 0 .1679842-0.000126 « 0.000150 +0. 000068 + 0.000004 = 0.000222. Approximate 95% c.l. were obtained by multiplying the standard deviation by 1.96, which resulted in the values n = 0.028 ± 0.029. From this 95% confidence interval is seen that the predicted 7t- value does not differ significantly from zero, which is consistent with the two-sided version of the ^-compari­son of frequencies of juvenile and adult carriers given above. By far the largest contribution to the variance of 7t derived from 0r Thus, in order to improve on the pre­cision of the calculated 7i-value, particularly the sample size for estimation of 0, should be increased.

Evaluation of bias by means of formula 4 showed that the second order partial derivative for 0j included a fac­tor of (1-(|>)2, while the corresponding term for 0a was (1- <}>)<|). This means that for species with high adult survival, like the pink-footed goose, uncertainty in 0a will make a much larger contribution to any bias, while, for example, for a species with an annual adult survival of 0.5 the two carrier percentages would contribute more equally. For the present data set, bias was calculated (using formula 4) to be b - 0.5 [0.077409 0.0002691 + 0.471963 0.000805 + 0.038221 0.000126] = 0.000297. This shows that the estimated value of n is positively biased, i.e. n is overes­timated, but if the number of X-rayed older geese is large,1. e. VAR(0a) is small, the calculated value of n is relative­ly unaffected.

The distribution of n was examined by Monte Carlo

N, N,

TT TT TTFigure 2. Distribution of 7t-values for 10,000 data sets generated by Monte-Carlo simulation, with samples sizes of 69 first-years and 286 older geese (left diagram). For the shown series, average observed frequencies were 0j = 0.244 and 0a = 0.358. The corresponding distribu­tions resulting from increasing the sample size of first-years to 286 (middle diagram) or from decreasing the sample size of older geese to 69 (right diagram) are given.

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simulations (Bootstrapping). For the observed sample sizes and carrier frequencies, the distribution of n is ap­proximately normal (Fig. 2). The variance of the simulat­ed distribution was V(it) = 0.000221, with 95% c.l. 7t = 0.028 ± 0.029, in good agreement with the results of for­mula 3.

The gain in precision that could be obtained from in­creasing the sample size of first-year geese was evaluat­ed by repeating the simulations for Nj = 286 (i.e. equal sample sizes of juveniles and adults). This would have re­sulted in a 50% reduction of the variance of 7t (to 0.000103), with corresponding 95% c.l. n = 0.028 ± 0.020, i.e. a 7t-value roughly between one and five per cent.

Furthermore, the loss of precision in 7t with small sam­ple sizes was evaluated by simulating data sets with Na = 69, i.e. by small samples of both first-year and older geese. This resulted in considerable loss of precision (see Fig. 2). The average value of K was 0.029, close to the ex­pected value of 0.028. The variance of this distribution, however, was V(7t) = 0.000409, with 95% c.l. n = 0.029 ± 0.040.

Discussion

Infliction rates and age of bagged geeseIn addition to the statistical evaluation of the model pre­sented above, the model’s performance can be assessed by comparison of the predicted infliction rates for the two age classes. The bag sizes of 1,000 first-year and 2,000 older geese are relatively well documented. It is not known whether the ratio of inflicted to bagged geese de­pends on age, but a plausible first hypothesis would be that this is not the case. This hypothesis would imply that pellets are inflicted upon roughly two older geese for each first-year individual. Alternatively, if the ratio of inflict­ed to bagged individuals depends on age, older geese might be predicted to have the higher ratio because 1) they are larger and heavier (J. Madsen, unpubl. data) and so should be physically more robust to shot, and 2) they are

more wary (as evidenced by the age composition of the bag being biased towards first-years), which could mean that the average distance at which they are shot at is larg­er, increasing the probability that they are wounded in­stead of being killed. For these reasons, the model was a priori expected to predict infliction of pellets upon at least two older geese for each first-year.

From the observed values of 0j (0.246) and 0a (0.360), the value of K calculated from the model was 0.028. As­suming that sizes of the two age classes are 4,000 first- years and 29,000 older geese, these rates correspond to infliction of pellets upon ca 1,000 first-years and 800 old­er geese annually (Table 2, A), or to 0.8 older goose for each first-year. In our opinion, therefore, the model failed to predict what might be considered realistic infliction rates for the two age classes.

Uncertainty in 7i could not explain this inconsistency; if pellets are inflicted upon 1,000 first-years annually, pel­lets should be inflicted upon 2,000 older geese, or 6.9%, a rate that clearly cannot be accounted for by the obser­vations (see Fig. 2). For 7t = 0.069, a frequency of adult carriers of ca 50% should be expected (see Fig. 1). Thus, the failure of the model to predict a realistic age distribu­tion is apparently caused by a 'too small' difference be­tween the observed frequencies: if 24.6% carry pellets af­ter one hunting season, 36.0% older carriers is too low a figure, given the high survival of the species and the age- composition of the bag.

Evidence in support of this conclusion was obtained from an investigation of bean geese Anserfabalis in Swe­den (Jonsson et al. 1985). For this species, 28% of the geese carried pellets after their first hunting season, while 62% of the older geese carried pellets. That difference was highly significant.

Two possible explanations for the inconsistency of the observed frequencies can be envisioned, viz. 1) the ob­served values of 0j and 0a are influenced by sampling vari­ation, and hence their true values might differ from the observed values, and 2) the model is wrong. In the fol­lowing, we discuss these possible explanations separate­ly-

Table 2. Assessment of annual numbers of first-year (Njuv) and adult (Nad) pink-footed geese inflicted with pellets and surviving (sexes pooled). Input values are frequency of adult (9a) and first-year (0j) pellet carriers as detected by X-ray photography, and adult survival rate <f). From these values, annual rates of infliction of adult individuals (7t) are calculated. The details of the various calculations (A-E) are giv­en in the text. Bag size is total bag for Denmark and Norway, and the age distribution is based on Danish wing surveys.

9a 6: 0 n Bag size N* Njuv

A 0.360 0.246 0.838 0.028 2,000 ad. + 1,000 juv. 810 1,020B 0.360 0.151 0.838 0.050 - 1,440 625C 0.407 0.246 0.838 0.042 - 1,210 1,020D 0.407 0.151 0.838 0.065 - 1,875 625E 0.300 0.151 0.859 0.029 " 835 625

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Sampling variation of carrier frequenciesThe sample size of first-year individuals was small, and 95% c.l. correspondingly large. Use of the lower 95% c.l. for 9j (0.151) led to an estimated value of 7t of 0.050, which would correspond to infliction of pellets upon 625 first-years and 1,440 older geese annually (see Table 2, B). This would be consistent with an age-independent ra­tio of inflicted to killed geese. Hence, the observed incon­sistency could be explained by assuming that the true per­centage of first-year carriers was close to the lower 95% confidence limit.

With respect to 0a, the upper 95% c.l. (0.407) corre­sponded to infliction of pellets upon ca 1,000 first-year and 1,200 older geese annually (see Table 2, C). Thus, be­cause of the larger sample sizes for older geese, variation in 9a could not account convincingly for the inconsisten­cy.

A combined overestimation of 0a and underestimation of 0; could more than explain the discrepancy. Three times as .many adults as juveniles are predicted to be in­flicted (see Table 2, D). Clearly, one way to account for the inconsistency was to assume that either one (first- year) or both observed carrier frequencies differed from the true values to an extent where they might be consid­ered unrepresentative. In this case, the 'bootstrapped' dis­tribution of 7t-values in Figure 2 would likewise be based on wrong assumptions. A more realistic distribution would then be obtained by shifting the shown distribution five intervals to the right.

Model assumptionsIn assuming population stability, no senescence, constant rates of infliction over years and individuals, and identi­cal survival rates of carriers and non-carriers, the model undoubtedly represents a simplified version of reality.

For a growing population, the proportion of recruits in the adult segment will be larger than assumed (i.e. >1 - <j)). Since 9j < 9a, violation of this assumption will lead to underestimation of 0a, and consequently to underestima­tion of K. However, the pink-footed goose population has been growing at an annual rate of only 1 % during the in­vestigated period (Madsen & Mitchell 1994). This rate is too low to affect the estimated 7t-value.

Effects of senescence were assessed by means of a trun­cated survivorship curve. For infliction rates of 0.246 and 0.069 (i.e. for age independent ratios of inflicted to bagged numbers), a maximum life-span of 9-10 years would be required to account for the observed 36% fre­quency of adult carriers. Ringing recoveries show that pink-footed geese reach considerably higher ages (e.g. 21 years, Cramp & Simmons 1977), hence this is clearly un­realistic.

The use of constant annual infliction rates was invoked

to represent an ‘average’ situation. Over the three years of this investigation, little variation in carrier frequencies was observed (see Table 1). Moreover, X-ray photogra­phy of 58 adult geese collected for physiological analy­sis (by rifle shooting) in Denmark and Norway, January- May 1996, revealed 21 pellet carriers, or 36.2% (J. Mad­sen, unpubl. data). This evidence clearly supported the as­sumptions of time constancy and representability of car­rier frequencies in cannon-net catches. Hence, there is no evidence that the assumptions of constant annual inflic­tion rates and carrier percentages are violated. The as­sumption that the probability of infliction was identical for all individuals admittedly could be violated, but if some subset of the geese had a higher probability of in­fliction they would undoubtedly also have a higher prob­ability of being killed by hunters, and in this case a low­er survival of pellet carriers might be expected.

The one untested assumption of the model thus ap­peared to be that survival of pellet carriers and non-car­riers was identical. Obviously, violation of this assump­tion by higher mortality of carriers could cause systemat­ic underrepresentation in X-ray sampling. To assess this possibility, the model was generalised to include diffe­rential survival of non-carriers ((|)n) and carriers ((|>c). In this case, mean adult mortality would be 1 - (l-0a)cf)n - 0a(f>c, and for a stable adult population this would be the fre­quency of recruits. Mean survival of adults and frequen­cy of recruits would thus depend on 0a, and possibly on n. After the second hunting season (i.e. at the time of X- raying), the expected frequency of non-carriers amongst the newly recruited adults would be

j _ 0 (l-O ,)(l-7t)0„(l-0;.)(l-jr)^+(0.+n(l-0J.))^

while the relative frequency of non-carriers in each fol­lowing age class would be

i _ e q-e„-,)q-JU4>„O-0a,-l)(1-")<l>n+(9<1,-l+^q-9a,-l)):<t>c

(i = 2,3,...).Using the relative size of the first age class (l-(l-0a)(|)n +

0acf>c) as a starting point, the size of each following age class can be calculated successively by

a, = a,,, [(1-0„.M)(1-Jt)4>„ + (0a,,+7t(l-0a, 1))<t)J

(i = 2,3,...). Note that in the case of identical survival rates of carriers and non-carriers, the expressions collapse into formula 1 given above.

Since overall survival was 0.838, (|>c and <(>„ were further constrained to the relationship 0.838 = 0a<f)c + (1 - 9a)4>n, i.e. the equation (|)c = (0.838 - (l-9a)<|>n)/0a must be satis­fied. Combining this and the formulas given above, the overall adult carrier frequency 9a can be found as in for-

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0, 6a

it

,/jrAT/ // /

/ /

- y4>n = 0.850 <j>c = 0.817

TT TT TTFigure 3. Predicted relationship between frequency of adult carriers (9a) and infliction rate (tt) if survival differs for non-carriers (<|)n) and carriers (<t>c). The frequency of pellet carriers among recruits is 0j = 0.246, annual adult infliction rate is tt = 0.069, and (|>n and <)>c are con­strained to a mean population survival of 0.838. Survival rates of 0.850 and 0.817 predict over 40% adult carriers, i.e. a higher than observed value (left diagram). Conversely, survival rates of 0.900 and 0.728 predict less than 30% adult carriers (right diagram). To predict the cor­rect frequency of 36.0% adult carriers, survival must be 0.871 for non-carriers and 0.779 for carriers (middle diagram). The graph for iden­tical survival of carriers and non-carriers ($„ = (|)c) is shown for comparison (thin dashed line, cp. Fig. 1), and tt and 0a are indicated by ver­tical and horizontal thin lines.

mula 1 above, by averaging over age classes. This was done numerically. The successive values of 0aa and a, cor­responding to 4>n were calculated, and the resulting value of 9a was found by summation over cohorts. This process was then repeated iteratively, until the value of (|>n predict­ing the correct value of 0a (0.36) had been determined.

Assuming that = 0.246 and that the ratio of bagged to wounded geese is similar for first-year and older geese, i.e. that ca 2,000 older geese are inflicted annually (Tt = 0.069), the model predicts that if the inconsistency is ex­plained by differential mortality between carriers and non-carriers, the values of <|>n and <j)c would have to be ap­proximately 0.871 and 0.779 (Fig. 3). It should be noted that for this more general model the assumption of no sampling covariance between 0a and (J) would be invali­dated, i.e. formulas 3 and 4 are not valid for the model with differential mortality.

Concluding remarksThe proposed model rests on rather strict assumptions with respect to population stability and constancy of an­nual survival and harvest rates. These assumptions may represent species with high annual survival rates - like the pink-footed goose - better than species with lower annu­al survival and presumably more fluctuating population sizes and harvest rates. Hence, the model should not be applied to data for species with fluctuating survival and harvest rates and population sizes unless sensitivity anal­ysis can demonstrate that the conclusions are robust.

The results of the statistical analysis of the model’s

properties, moreover, emphasised that this approach should only be used when numbers of X-rayed individ­uals are sufficient to ensure a fairly high precision of the estimated carrier frequencies. For the pink-footed geese, the major source of uncertainty undoubtedly is that only 69 first-year individuals were X-rayed, but in addition the estimated annual infliction rate n will be biased unless sample sizes of older birds are substantial.

Given these limitations, we found this model useful in the initial analyses of the pink-footed goose data. In the first step, the model showed that relatively small annual infliction rates - below ca 0.07 - could account for the ob­served frequency of adult pellet carriers; in the next step, the model helped us to realise that the predicted infliction rates were inconsistent with the age distribution of bagged geese, and that the observed frequencies of pellet carriers in the two age classes were mutually conflicting. The pos­sibility of random variation affecting the observed fre­quencies could not be excluded, but in that case the true value of 0j would have to be considerably lower than the observed one.

Assuming that the first of these explanations was true, estimates of minimum infliction rates were obtained by combining the lower 95% c.l. of carrier frequencies for both first-year and older geese with the upper 95% c.l. of survival. The resulting infliction rate was 7t = 0.029, cor­responding to infliction of pellets to 625 first-year and 835 older geese annually (see Table 2, E). Though these minimum numbers tend to be inconsistent with the age distribution expected on the basis of bag composition, they cannot be directly refuted.

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Compared to the overall bag size of ca 3,000 geese, the estimated minimum of pellets being inflicted upon 1,460 geese annually thus suggests that the ratio of inflicted to bagged geese is at least ca 0.5. To derive this minimum estimate, moreover, we used the most conservative as­sumptions of sampling variation in both 0j, 0a and <|>, which together represent an unlikely combination.

In addition to the geese identified as carriers by X-ray examination, an unknown loss of crippled, non-retrieved individuals must be added, together with the geese which are wounded but survive without retaining pellets. Few data exist concerning the size of these proportions, but for North American mallards Anas platyrhynchos Bellrose (1953) presented data suggesting that they may add a fur­ther 0.5-1.0 individuals for each individual inflicted with pellets. Thus, the minimum ratio of wounded to bagged individuals is likely to be substantially higher than 0.5.

Alternatively, the discrepancy between the 24.6% first- years and the 36.0% older geese carrying pellets might be explained by lower survival of pellet carriers. In this case, the estimated annual infliction rates will be somewhat higher. The 'differential survival' hypothesis can be test­ed by analysis of resightings of neck-banded geese. This will be treated in a companion paper (Madsen & Noer 1996).

Acknowledgements - several of our colleagues, in particular Ebbe Bpgebjerg, helped and cooperated in the field. The staff at Santax AJS was very helpful. Anthony D. Fox and Hugh Boyd commented on an earlier version of the MS, and Nigel G. Yoccoz and two anon­ymous referees provided many helpful suggestions. The study was supported by grants to Jesper Madsen from the Danish Research Councils (Grant no 11-9534-1 and no 11-0328-1).

ReferencesAnkney, C.D. 1975: Incidence and size of lead shot in lesser snow

geese. - Wildlife Society Bulletin 3(1): 25-26.Anon. 1970: Mathematics and statistics. - Geigy Scientific Tables,

7th ed. Ciba-Geigy ltd. Basle, Switzerland, 198 pp.Bellrose, F.C. 1953: A preliminary evaluation of cripple losses in

waterfowl. - Transactions 18th North American Wildlife Confe­rence: 337-360.

Clausager, I. 1990: Vingeindsamling frajagtstesonen 1989/90 i Dan­mark. - National Environmental Research Institute. Technical Re­port from NERI No. 1, 39 pp. (In Danish with English summary).

Clausager,1.1991: Vingeindsamling frajagtsaesonen 1990/91 iDan- mark. - National Environmental Research Institute. NERI Tech­nical Report No. 31,58 pp. (In Danish with English summary).

Clausager,1.1992: Vingeindsamling frajagtsaesonen 1991/92iDan- mark. - National Environmental Research Institute. NERI Tech­nical Report No. 58, 53 pp. (In Danish with English summary).

Clausager, 1.1993: Vingeindsamling frajagtsaesonen 1992/93 i Dan­mark. - National Environmental Research Institute. NERI Tech­nical Report No. 85, 58 pp. (In Danish with English summary).

Clausager, I. 1994: Vingeindsamling frajagtsaesonen 1993/94iDan- mark. - National Environmental Research Institute. NERI Tech­nical Report No. 115, 52 pp. (In Danish with English summary).

Cramp, S. & Simmons, K.E.L. 1977: Handbook of the Birds of Eu­rope, the Middle East, and North Africa. Vol. I. Ostrich to ducks. - Oxford University Press, 722 pp.

Efron, B. 1979: Bootstrap methods: another look at the Jackknife. - The Annals of Statistics 7(1): 1-26.

Elder, W.H. 1955a: Fluoroscopic measures of shooting pressure on Pink-footed and Greylag Geese. - Wildfowl Trust Annual Report 7: 123-126.

Elder, W.H. 1955b: Fluoroscopic measures of hunting pressure in Europe and North America. - Transactions 20th North American Wildlife Conference: 298-321.

Grieb, J.R. 1970: The shortgrass prairie Canada goose population. - Wildlife Monographs 22, 47 pp.

Jonsson, B., Karlsson, J. & Svensson, S. 1985: Incidence of lead shot in tissues of the bean goose (Anser fabalis) wintering in South Sweden. - Swedish Wildlife Research 13(3): 259-271.

Madsen, J. 1987: Status and management of goose populations in Europe, with special reference to populations resting and breed­ing in Denmark. - Danish Review of Game Biology 12(4): 1-74.

Madsen, J. & Mitchell, C. 1994: Status of the pink-footed goose, 1990-1993. - IWRB Goose Research Group Bulletin 5: 8-11.

Madsen, J., Asferg, T., Clausager, I. & Noer, H. 1996: Status og jagt- tider for danske vildtarter. - Temarapport 1996/6, National Envi­ronmental Research Institute, 112 pp. (In Danish).

Madsen, J. & Noer, H. 1996: Decreased survival of pink-footed geese Anser brachyrhynchus carrying shotgun pellets. - Wildlife Biology 2: 75-82.

Seber, G.A.F. 1982: The estimation of animal abundance and relat­ed parameters. - C. Griffin & Co., London and High Wycombe (2nd ed.), 654 pp.

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